A golden rectangle is a rectangle with ratio [math]\frac{length}{width}=\Phi[/math].[br]A real golden spiral isn't a spiral consisting of quarter circles with increasing radius, but a [url=https://en.wikipedia.org/wiki/Logarithmic_spiral]logaritmic spiral[/url]. In such a spiral every 90° turn the distance between the pole and the points on the spiral increases wih factor [math]\Phi[/math].[br]Starting from some nice properties you can combine such a logaritmic spiral with a golden rectangle. Even if you are not familiar with parametric curves, you can follow these properties in following applet.[br][br][list][*]You can find all points in which the tangents to the curve are vertical as the intersection points of the line through the pole P with slope the arctangens of the parameter b.[/*][*]You can find the points in which the tangents are horizontal on the line through P, perpendicular to the first line.[/*][*]You can control that the radii at the consecutive intersection points with the spiral indeed are increasing with factor [math]\Phi[/math] every 90°.[/*][*]Drawing the tangents in four consecutive tangent points you get a golden rectangle.[/*][/list]Note: [br][list][*]The 4th point (below left), in which the tangent is vertical, doesn't lie in this golden rectangle. [/*][*]If you draw a spiral using circular arcs in this golden rectangle, you notice that this approaches the real logaritmic spiral. In this spiral the point below left (with the biggest radius) is a corner of the golden rectangle. Always at the start of a new circular arc the distortion is clearly visible.[/*][/list]